Multipole Expansion

where the

(340) |

are known as the

The most important
are those corresponding to
,
, and
, which are known as *monopole*, *dipole*,
and *quadrupole* moments, respectively. For each
, the multipole moments
, for
to
, form an
th-rank tensor with
components.
However, Equation (310) implies that

(341) |

Hence, only of these components are independent.

For , there is only one monopole moment. Namely,

(342) |

where is the net charge contained in the distribution, and use has been made of Equation (312). It follows from Equation (340) that, at sufficiently large , the charge distribution acts like a point charge situated at the origin. That is,

(343) |

By analogy with Equation (195), the dipole moment of the charge distribution is written

(344) |

The three Cartesian components of this vector are

(345) | ||

(346) | ||

(347) |

On the other hand, the spherical components of the dipole moment take the form

(348) | ||

(349) | ||

(350) |

where use has been made of Equations (313)-(315). It can be seen that the three spherical dipole moments are independent linear combinations of the three Cartesian moments. The potential associated with the dipole moment is

(351) |

However, from Equations (313)-(315),

(352) | ||

(353) | ||

(354) |

Hence,

(355) |

in accordance with Equation (200). Note, finally, that if the net charge, , contained in the distributions is non-zero then it is always possible to choose the origin of the coordinate system in such a manner that .

The Cartesian components of the *quadrupole tensor* are defined

(356) |

for , , , . Here, , , and . Incidentally, because the quadrupole tensor is symmetric (i.e., ) and traceless (i.e., ), it only possesses five independent Cartesian components. The five spherical components of the quadrupole tensor take the form

(357) | ||

(358) | ||

(359) | ||

(360) | ||

(361) |

Moreover, the potential associated with the quadrupole tensor is

(362) |

It follows, from the previous analysis, that the first three terms in the multipole expansion, (340), can be written

(363) |

Moreover, at sufficiently large , these are always the dominant terms in the expansion.